Quantum Finite Automata and Logic

Transcription

1 University of Latvia Ilze Dzelme-Bērziņa Quantum Finite Automata and Logic Doctoral Thesis Area: Computer Science Sub-Area: Mathematical Foundations of Computer Science Scientific Advisor: Dr. habil. math., Prof. Rūsiņš Freivalds Riga 2010

2 Abstract The connection between the classical computation and mathematical logic has had a great impact in the computer science which is the main reason for the interest in the connection between the quantum computation and mathematical logic. The thesis studies a connection between quantum finite state automata and logic. The main research area is a quantum finite state automaton and its different notations (measure-once quantum finite state automaton, measure-many quantum finite state automaton, and Latvian quantum finite state automaton), more precisely, the languages accepted by the various models of the quantum finite state automaton and its connection to languages described by the different kinds of logic (first order, modular etc.). Additionally, a quantum finite state automaton over infinite words is introduced. The first part of the thesis is devoted to the connection between such quantum finite state automata as measure-once quantum finite state automata, measure-many quantum finite state automata, and Latvian quantum finite state automata and first order logic, modular logic, and generalized quantifiers - Lindström quantifier and group quantifier. For measure-once quantum finite state automata, we have characterized the language class accepted by measure-once quantum finite state automata in terms of logic using generalized quantifiers - Lindström quantifier and group quantifier, studied the relationship between the language class accepted by measure-once quantum finite state automata and the language class described by first order logic and modular logic. For measure-many quantum finite state automata, the connection between language classes accepted by quantum finite state automata and first order logic and modular logics has been studied, as well as the connection between acceptance probability of quantum finite state automata and logic. We also examined the language class accepted by Latvian quantum finite state automata in terms of logic. The second part is devoted to the quantum finite state automata over infinite words. We extend the notation of quantum finite automata for infinite words. The class of languages accepted by Büchi quantum finite state automata has been studied and we examine the closure properties of Büchi quantum finite state automata.

3 Anotācija Matemātiskās loǧikas un klasiskās skaitļošanas saistībai ir bijusi liela nozīme datorzinātnes attīstībā. Tas ir galvenais iemesls, kas raisījis interesi pētīt kvantu skaitļošanas un loǧikas saistību. Promocijas darbs aplūko saistību starp galīgiem kvantu automātiem un loǧiku. Pamatā pētījumi balstās uz galīgu kvantu automātu un tā dažādiem veidiem (galīgu kvantu automātu ar mērījumu beigās, galīgu kvantu automātu ar mērījumu katrā solī, galīgo "latviešu" kvantu automātu), precīzāk, valodām, ko akceptē dažādie kvantu automātu modeļi, un to saistību ar valodām, ko apraksta dažādie loǧikas veidi ( pirmās kārtas loǧika, modulārā loǧika u.c.). Darbā ir arī aplūkoti galīgi kvantu automāti, kas akceptē bezgalīgus vārdus. Promocijas darba pirmā daļa ir veltīta galīga kvantu automāta ar mērījumu beigās, galīga kvantu automāta ar mērījumu katrā solī un galīgā "latviešu" kvantu automāta saistībai ar pirmās kārtas loǧiku, modulāro loǧiku un loǧiku, kas izmanto vispārinātus kvantorus - Lindstroma kvantoru un grupas kvantoru. Galīgiem kvantu automātiem ar mērījumu beigās ir aprakstīta valodu klase, ko tie atpazīst, izmantojot vispārinātus kvantorus - Lindstroma kvantoru un grupas kvantoru, kā arī apskatīta galīga kvantu automāta ar mērījumu beigās saistība ar pirmās kārtas loǧiku un modulāro loǧiku. Galīgiem kvantu automātiem ar mērījumu katrā solī ir apskatīta to saistība ar pirmās kārtas loǧiku un modulāro loǧiku ne tikai no valodas atpazīšanas viedokļa, bet arī no galīga kvantu automāta ar mērījumu katrā solī akceptēšanas varbūtības viedokļa. Darbā aplūkota arī galīgā "latviešu" kvantu automāta saistība ar pirmās kārtas loǧiku un modulāro loǧiku, un aprakstīta valodu klase, izmantojot grupas kvantoru. Otrā darba daļa ir veltīta kvantu automātiem bezgalīgiem vārdiem. Autors paplašina kvantu galīgā automāta definīciju bezgalīgiem vārdiem. Darbā aplūko valodu klasi, ko atpazīst Büchi galīgs kvantu automāts, kā arī Büchi galīga kvantu automāta slēgtību pret apvienojumu, šķēlumu un papildinājumu.

4 Preface This thesis assembles the research performed by the author and reflected in the following publications: 1. Ilze Dzelme-Bērziņa. Quantum Finite State Automata over Infinite Words. UC2010: 9th International Conference on Unconventional Computation, Lecture Notes in Computer Science, vol 6079, pp , Ilze Dzelme-Bērziņa. Mathematical logic and quantum finite state automata. Theoretical Computer Science vol 410(20): Quantum and Probabilistic Automata, pp , Ilze Dzelme-Bērziņa. First Order Logic and Acceptance Probability of Quantum Finite State Automata. DLT th International Conference on Developments in Language Theory, Satellite Workshop on Probabilistic and Quantum Automata. Proceedings., TUCS General Publication No 45, pp , Ilze Dzelme. Quantum Finite Automata and Logics. SOFSEM 2006: Theory and Practice of Computer Science, 32nd Conference on Current Trends in Theory and Practice of Computer Science, Lecture Notes in Computer Science, vol 3831, pp , Ilze Dzelme-Bērziņa. Modular Logic and Quantum Finite State Automata. Proceedings of the Asian Conference on Quantum Information Science 2006, pp , Ilze Dzelme-Bērziņa. Formulas of first order logic and quantum finite automata. Proceedings of 8th International conference on Quantum Communication, Measurement and Computing, pp , NICT Press, Japan, Laura Mancinska, Maris Ozols, Ilze Dzelme-Berzina, Rubens Agadzanjans, Ansis Rosmanis. Principles of Optimal Probabilistic Decision Tree Construction.

5 International Conference on Foundations of Computer Science, CSREA Press, pp , Ansis Rosmanis, Ilze Dzelme-Berzina. Mixed States in Quantum Cryptography. International Conference on Foundations of Computer Science, CSREA Press, pp , The results of the thesis were presented at the following international conferences and workshops: 1. 13th Workshop on Quantum Information Processing QIP 2010, , Zürich, Switzerland. Poster Latvian Quantum Finite State Automata and Logic th International Conference on Developments in Language Theory. Satellite Workshop on Probabilistic and Quantum Automata (DLT 07), , Turku, Finland. Presentation First Order Logic and Acceptance Probability of Quantum Finite State Automata. 3. 8th International Conference on Quantum Communication, Measurement and Computing (QCMC 06), , Tsukuba, Japan. Poster Formulas of first order logic and quantum finite automata. 4. Asian Conference on Quantum Information Science 2006 (AQIS 06), , Beijing, China. Poster Modular Logic and Quantum Finite State Automata. 5. Theory and Practice of Computer Science, 32nd Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2006), , Merin, the Czech Republic. Presentation Quantum Finite Automata and Logics. Many people have contributed to the successful completion of the thesis. My gratitude to my supervisor, professor Rūsiņš Freivalds, who encouraged to start the research and whose support helped me to complete the work. Many thanks to Andris Ambainis for his support and helpfulness. Finally, my gratitude to the colleagues, friends, and relatives, especially to my daughter, husband, parents, and sister whose support has encouraged me and without whom the thesis would not be finished.

6 Contents Abstract Anotācija Preface i ii iii 1 Introduction Quantum Computation Quantum Turing Machine Quantum Automata Logics and Classical Computation Automata over Infinite Words Preliminaries Probabilistic Systems Quantum Mechanics Superposition Hilbert Space Observables and Measurement Unitary Evolution Mixed States and Density Matrices Quantum Finite State Automata Models of Classical Finite State Automata Measure-Once Quantum Finite State Automata Measure-Many Quantum Finite State Automata Latvian Quantum Finite State Automata Monoids Logic First Order Languages Modular Logic Generalized Quantifier Finite State Automata over Infinite Words Classical Automata over Infinite Words

7 3 Measure-Once Quantum Finite State Automata and Logic Measure-Once Quantum Finite State Automata and First Order Logic Measure-Once Quantum Finite State Automata and Modular Logic Measure-Once Quantum Finite State Automata and Generalized Quantifiers 45 4 Measure-Many Quantum Finite State Automata and Logic Measure-Many Quantum Finite State Automata and First Order Logic Measure-many quantum finite state automata and modular logic Latvian Quantum Finite State Automata and Logic Latvian Quantum Finite State Automata and First Order Logic Latvian Quantum Finite State Automata and Modular Logic Latvian Quantum Finite State Automata and Generalized Quantifiers Quantum Automata over Infinite Words Definition of Quantum Finite State Automata over Infinite Words Group Automata over Infinite Words Quantum Finite State Automata over Infinite Words with Büchi Acceptance Condition Closure Properties of Quantum Finite State Automata over Infinite Words with Büchi Acceptance Condition Measure-Many Quantum Finite State Automata over Infinite Words Conclusion 76 Bibliography 78

8 Chapter 1 Introduction The rapid development of the quantum computation and the huge impact of mathematical logic in the classical computation were the main reasons to study the relationship between quantum finite state automata and mathematical logic. The connection between logic and the classical automata theory started with the work of Büchi [17] and Elgot [22]. They showed how a logical monadic second-order formula can effectively be transformed into a finite state automaton accepting the language defined by the formula and vice versus - how a finite state automaton can be transformed to a logical monadic second order formula which specifies the language accepted by the automaton. The logical description of the computation models behaviour also influenced complexity theory. In 1974, Fagin [23] gave a characterization of non-deterministic polynomial time (NP) as the set of properties expressible in the second order existential logic. The above results inspired us to study the connection between the quantum automata theory and logic. The goal of the research was to describe language classes recognized by different quantum automaton models using logical formulas, to find properties of quantum automata that can be connected to logics. We have achieved the following: characterized the language class accepted by measure-once quantum finite state automata with bounded error in the terms of logic; proved that intersection of the language class accepted by measure-once quantum finite automata with bounded error and languages defined by F O[<] contains only trivial languages, i.e., an empty language and Σ ; proved that languages described by modular logic using only modular quantifiers are recognized by measure-once quantum finite state automata;

9 2 studied the connection between languages accepted by measure-many quantum finite state automata and first order logic, as well as, the connection between acceptance probability of measure-many quantum finite state automata and first order logic was examined; studied the connection between acceptance probability of measure-many quantum finite state automata and modular logic using first order quantifiers; studied the connection between Latvian quantum finite state automata and logic. The theory of classical automata over infinite strings has been applied in the various research areas, including the verification of reactive systems, reasoning about infinite games, and decision problems for certain logics. Our research of the connection between finite state automata and logic lead us to automata over infinite words. We devoted the second part of the thesis to quantum finite state automata over infinite words. We have achieved the following: extended definitions of group automata and measure-once quantum finite state automata to infinite words; defined Büchi, Streett, and Rabin acceptance conditions for quantum case; proved that our Büchi quantum finite state automata over infinite words with bounded error recognize the limit of the group languages; proved that our Büchi quantum finite state automata is closed under union; showed that our Büchi quantum finite state automata is not closed under intersection and complementation; defined a measure-many quantum finite state automaton over infinite input. In the Chapter 1, the background to the quantum computation and to the connection between mathematical logic and classical computation is presented. We give a brief history of development of the quantum computation, quantum Turing machine, and quantum automata. The section "Logics and classical computation" is devoted to the main results in the connection between classical computation and logic. And an overview of automata over infinite words is presented in the section "Automata over infinite words". The Chapter 2 contains main notations and definitions used in the thesis - probabilistic systems, brief introduction to quantum mechanics, notations of quantum finite state

10 1.1 Quantum Computation 3 automata used in the thesis, a brief overview to mathematical logic, algebra, and classical computation, as well as, introduction to automata over infinite words. The Chapter 3 is devoted to connection between measure-once quantum finite state automata and logic - first order logic, modular logic, and logic using generalized quantifiers - Lindström quantifier and group quantifier. The connection between measuremany quantum finite state automata and first order logic and modular logic is studied in the Chapter 4. In the Chapter 5, we study Latvian quantum finite state automata and its connection to logic. The Chapter 6 is devoted to the quantum finite state automata over infinite words, where we give a definition of quantum finite state automata over infinite words and obtained results. 1.1 Quantum Computation Over the past half century, the power of computers has doubled every year and a half. This phenomena is known as "Moore s law", named after Gordon Moore, who had stated the exponential advance in the 1960 s [39]. If the Moore s law is to be sustained then we must learn to build a computer based on the quantum physics (quantum computers represent the ultimate level of miniaturization) and study quantum computation. Quantum computation investigates the computation power and other properties of the computers based on quantum mechanic principles. The concept of quantum computing dates back to the early 80 s, to the speech of Nobel Prize winner Richard P. Feynman and accompanying paper "Simulating physics with computers" [24]. His paper was the first work that explicitly discussed the construction of machine operating according to the laws of quantum physics. Feynman discussed the idea of a universal simulator - a machine using quantum effects to explore other quantum effects and run simulations. In 1982 [13], Benioff described a quantum mechanical computation model. However, his model did not use any quantum mechanical effects, it was a hybrid Turing machine storing qubits on the tape instead of classical bits and measuring each qubit from the tape at each step. A year later in [3], David Albert described a self measuring quantum automaton that performed tasks which no classical computer could simulate. However, the machine was largely unspecified. By instructing the automaton to measure itself, it can obtain subjective information that is absolutely inaccessible by measurement from the outside. Finally in 1985 [20], Deutsch introduced a fully quantum computational model and gave the description of a universal quantum computer. Later [14], Bernstein and Vazirani introduced the construction of a universal quantum Turing machine capable

11 1.1 Quantum Computation 4 of simulating any other quantum Turing machine with polynomial efficiency. After the Deutsch paper [20] in the following years, there was a small interest in quantum computation. However during these years, one of the first examples of a quantum algorithm (The Deutsch - Jozsa algorithm [21]) that is more efficient than any possible classical algorithm has been presented, and Dan Simon [50] invented an oracle problem for which a quantum computer would be exponentially faster than a conventional computer. The main ideas of the algorithm were later developed in Shor s factoring algorithm [49]. Interest in the quantum computation increased dramatically, when in 1994 Peter Shor discovered efficient quantum algorithms for the problems of integer factorization and discrete logarithms [49]. The importance of the Shor s algorithm for finding factors is in fact that reliability of the widely used public-key cryptography schema RSA is based on the assumption that factoring large numbers is computationally infeasible. Shor demonstrated that it is not the case if we could build a quantum computer. Shor s results are the powerful evidence, that quantum computers are more powerful than Turing machines, even probabilistic Turing machines. Further evidence of the quantum automata s power came in 1996 when Grover showed, that the problem of conducting a search through some unstructured search space could be sped up by a quantum computer [26]. Although Grover s algorithm did not had as powerful speed-up as Shor s algorithm, the widespread applications of search-based methodologies has excited considerable interest in the algorithm of Grover. However, the theory of quantum computers is far more developed than the practice: a large scale quantum computer has not been built yet. In 1998, the first working 2-qubit Nuclear magnetic resonance (NMR) quantum computer was demonstrated by Jonathan A. Jones and Michele Mosca at Oxford University. In the same year, the first working 3- qubit quantum computer has been developed and the first execution of Grover s algorithm on an NMR computer has been performed. In 2006, the theorists and experimentalists at the Institute for Quantum Computing and Perimeter Institute for Theoretical Physics in Waterloo, along with MIT, Cambridge, have presented an operational control method in quantum information processing extending up to 12 qubits. The more developed field of quantum theory is quantum cryptography. Although large scale quantum computers are not built, the quantum devices for cryptography have been provided. Quantum cryptography was first proposed by Stephen Wiesner, who, in the early 1970s, introduced the concept of quantum conjugate coding. His paper "Conjugate Coding" was rejected by IEEE Information Theory, but it was eventually published in 1983 in SIGACT News (15:1 pp , 1983). In the early 1980s, Charles H. Bennett

12 1.1 Quantum Computation 5 and Gilles Brassard proposed a method for secure communication based on Wiesner s "conjugate observables". In 1990, independently and initially unaware of the earlier work, Artur Ekert developed a different approach to quantum cryptography based on peculiar quantum correlations known as quantum entanglement. Quantum cryptography is also used in practice. Quantum encryption technology provided by the Swiss company Id Quantique was used in the Swiss canton of Geneva to transmit ballot results to the capitol in the national election in In 2004, the world s first bank transfer using quantum cryptography was carried out in Austria. An important cheque, which needed absolute security, was transmitted from the Mayor of the city to an Austrian bank. The world s first computer network SECOQC (Secure Communication Based on Quantum Cryptography) protected by quantum cryptography was implemented in October 2008, at a scientific conference in Vienna. The network used 200 km of standard fibre optic cable to interconnect six locations across Vienna and the town of St Poelten located 69 km to the west. For more information about quantum computation please refer to [28], [27], and [41], but now we are going to consider the main models of quantum computation - a quantum Turing machine and a quantum automaton. In comparison with quantum Turing machine, a quantum automaton has finite memory and the computation steps does not exceed the length of the input Quantum Turing Machine The beginnings of the modern computer science goes back to a remarkable paper [54] of Alan Turing, known as father of modern computer science, written in He developed an abstract notation of computation now known as Turing machine. Turing showed that there is a Universal Turing machine that can be used to simulate any other Turing machine. The Turing machine later evolved into the modern computer. Quantum computation also has its Turing machine - quantum Turing machine. Quantum Turing machine is an abstract machine which is used to model the effect of the quantum computation. Any quantum algorithm can be expressed using a particular quantum Turing machine. We may say that quantum Turing machines have the same relation to the quantum computation as Turing machines have in the classical computation. As already mentioned, the first quantum computational model (quantum Turing machine)

13 1.1 Quantum Computation 6 was proposed by Deutsch [20]. Afterwards [14], Bernstein and Vazirani introduced the construction of a universal quantum Turing machine capable of simulating any other quantum Turing machine with polynomial efficiency. However, quantum Turing machines are not always used for the quantum computation analysis, in quantum information theory a quantum circuit is more commonly used model. It has been proved that quantum Turing machines and quantum circuits are computationally equivalent. [19] Iriyama, Ohya, and Volovich have developed a model of a Linear Quantum Turing Machine [31]. This is a generalization of a classical quantum Turing machine that has mixed states and that allows irreversible transition functions. The quantum Turing machine with postselection was defined by Scott Aaronson, who showed that the class of polynomial time on such a machine is equal to the classical complexity class PP (the class of decision problems solvable by a probabilistic Turing machine in polynomial time) [2]. Another model of quantum Turing machine is the classically-controlled quantum Turing machine - a Turing machine with a quantum tape for acting on quantum data, and a classical transition function for a formalized classical control was introduced by Perdrix and Jorrand in [43], where they showed that any classical Turing machine can be simulated by a classically-controlled quantum Turing machine without loss of efficiency Quantum Automata A natural model of classical computing with finite memory is a finite state automaton, likewise a quantum finite state automaton is a natural model of quantum computation. Quantum finite state automata refer to the quantum computers in a similar way as finite state automata are related to Turing machines. An automaton reads input symbols from the given input and performs a transition function defined for the input symbol, after the input is read, the automaton accepts or rejects an input word. A quantum automaton can reject or accept a word with a probability between zero and one. Different notations of quantum finite state automata are used. The most simple and one of the most popular notation of quantum finite state automata is a definition of a quantum finite state automaton introduced by Moore and Crutchfield [38] known as measure-once quantum finite state automaton (MO-QFA). The measure-once quantum finite state automata performs unitary transformation for each input symbol and makes the only measurement when the whole word is read obtaining the result whether the input is accepted or rejected. MO-QFA is pure state model of quantum finite state automata. Brodsky and Pippenger have proved [16], that MO-QFA with bounded error recognize the same language class as group automata [52]. In the same paper [16], Brodsky and

14 1.1 Quantum Computation 7 x x qi y qj Figure 1.1 The forbidden construction of [16]. Pippenger showed that measure-once quantum finite state automaton can be simulated by a probabilistic finite state automaton and described an algorithm that determines if two measure-once quantum finite state automata are equivalent. However, if we consider the measure-once quantum finite state automata with unbounded error it can recognize non-regular languages, for example, L 1 = {ω ω {0, 1}, ω 0 = ω 1 }. Another widely used notation of quantum finite state automata is a measure-many quantum finite state automaton (MM-QFA) introduced by Kondacs and Watrous [32]. MO-QFA and MM-QFA have seemingly small difference, the definition of measure-once quantum finite state automata allows the measurement only at the end of the computation, but the definition of measure-many quantum finite state automata allows the measurement at every step of the computation (the measurement provides a probabilistic decision on every input symbol by projecting a state on the subspace of accepting states, the subspace of rejecting states, and the subspace of the automaton s non-halting states). The computation of MM-QFA halts when an accepting state or a rejecting state is reached. While a measure-many quantum finite state automaton is in a non-halting state, the computation continues. Although MM-QFA is more powerful than MO-QFA, measuremany quantum finite state automata with bounded error recognise only the subset of the regular languages. The several attempts have been made to define the language class of measure-many quantum finite state automata. Brodsky and Pippenger [16] introduced the first forbidden construction of MM-QFA (see the figure 1.1) - if a minimal deterministic finite state automaton of a language contains such construction then the language cannot be recognized by a measure-many quantum finite state automaton. Later Ambainis, Ķikusts and Valdats [7] have shown another forbidden construction (the figure 1.2) for measure-many quantum finite state automata, but it is still an open question how to characterize the language class accepted by measure-many quantum finite state automata. In the thesis, we also consider a notation of Latvian quantum finite automata which was introduced in [1]. A Latvian quantum finite state automaton uses mixed states, at every step of the computation an automaton performs a unitary transformation and a projection defined for each input symbol. It has been provided an algebraic characterization of the languages recognized by Latvian quantum finite state automata. Enhanced quantum finite state automata is the measure-many version of Latvian quantum

15 1.2 Logics and Classical Computation 8 x y x y x y z 1 z 2 z 2 z 1 Figure 1.2 The forbidden construction of [7]. finite state automata, it was defined in [8], where it was shown that there are languages for which enhanced quantum finite automata take exponentially more states than those of the corresponding classical automata. There are also other notations of quantum finite state automata as: one-way quantum finite automata with control language [15] - the accepting behaviour is controlled by the result of the projective measurement performed at each step in the computation, it was proved that one-way quantum finite automata with control language with bounded error recognize exactly regular languages [36]; one-way quantum finite automata together with classical states with bounded error accepting all regular languages [46]; ancilla quantum finite state automata, where an ancilla quantum part is imported, and then the internal control states and the states of the ancilla part together evolve by a unitary transformation [42]; measure-once one-way general quantum finite state automata and measure-many one-way general quantum finite state automata [33] instead of a unitary transformation a trace-preserving quantum operation is used. 1.2 Logics and Classical Computation The connection between automata theory and logic dates back to the early sixties to the work of Büchi [17] and Elgot [22], who showed that the finite automata and monadic second order logic (interpreted over finite words) have the same expressive power, and that the transformation from logical monadic second order formulas to finite state automata and vice versus are effective. Later, the equivalence between finite state automata and monadic second order logic over infinite words and trees were shown in the works of

16 1.3 Automata over Infinite Words 9 Büchi [18], McNaughton [35], and Rabin [47]. The reduction of formulas to finite state automata was the key to prove decidability results in mathematical theories. The next important step in the connection between automata theory and logic was Pnueli s work [45]. It was proposed to use temporal logic for reasoning about continuously operating concurrent programs. In the eighties, temporal logics and fixedpoint logics took the role of specification languages and more efficient transformations from logic formulas to automata were found. This led to powerful algorithms and software systems for the verification of finite state programs ( model-checking"). The research of the equivalence between automata theory and logic formalism also influenced language theory itself. For example, the logical approach helped generalizing the domain of words to more general structures like trees and partial orders. The logical description of the computation models behaviour also influenced complexity theory. In 1974, Fagin [23] gave a characterization of non-deterministic polynomial time as the set of properties expressible in the second order existential logic. Later, Immerman [29] and Vardi [55] characterized polynomial time as the set of properties expressible in the first order inductive definition, which is defined by adding the least point operator to the first order logic. In the similar way, polynomial space has also been characterized [30] as second order logic with transitive closure. These results led to the development of a new field - Description complexity - a sub field of computational complexity theory and mathematical logic, which seeks to characterize complexity classes by the type of logic needed to express the language in them. A merge of techniques and results from automata theory, logic, and complexity was achieved in circuit complexity theory, which studies the computational power of boolean circuits, regarding restrictions in their size, depth, and types of allowed gates. Natural families of circuits can be described by generalized models of finite state automata as well as by appropriate systems of the first order logic. 1.3 Automata over Infinite Words The study of finite state automata working on infinite words was initiated by Büchi [17]. Büchi discovered connection between formulas of the monadic second order logic of infinite sequences (S1S) and ω-regular languages, the class of languages over infinite words accepted by finite state automata. The complexity of theory of automata over infinite words was evident from the inital work of Büchi, where he showed that nondeterministic automata over infinite words are strictly more powerful than deterministic automata over infinite words.

17 1.3 Automata over Infinite Words 10 Few years later after Büchi paper, Muller proposed an alternative definition of finite automata on infinite words [40]. McNaughton proved that with Muller s definition, deterministic automata recognizes all ω-regular languages [35]. Later, Rabin extended decidability result of Büchi for S1S to the monadic second order of the infinite binary tree (S2S) [47]. Rabin s theorem can be used to settle a number of decision problems in logic. A theory of automata over infinite words has started from these studies. It can be applied in the various research areas, including the verification of reactive systems, reasoning about infinite games, and decision problems for certain logics. Recently probabilistic variants of finite state automata over infinite words have been introduced and studied in [12], [10], and [11].

18 Chapter 2 Preliminaries The chapter provides the main notations, definitions, and results of the quantum computation and connection between logic and classical computation which are going to be helpful for the rest of the thesis. Additionally, we give definitions of the classical finite state automata over infinite words. The most of the definitions we refer to [28], [27], and [41] for Quantum Computation. For the main results and definitions of the connection between logic and automata and automata over infinite words, we refer to [25] and [53]. 2.1 Probabilistic Systems A system admitting probabilistic nature means that we do not know for certain the state of the system, but we know that the system is in the states x 1,..., x n with probabilities p 1,..., p n that sum up to 1. Definition Notation p 1 [x 1 ] + p 2 [x 2 ] p n [x n ] (2.1) where p i 0 and p 1 + p p n = 1 stands for a probability distribution, meaning that the system is in state x i with probability p i. We also call distribution 2.1 a mixed state. States x i are called pure states. Example Tossing a fair coin will give head h or tail t with a probability of 1 2. The notation 1h + 1 t is the mixed state of a fair coin tossing. 2 2 Instead of dealing with only one possible "reality" of how the process might evolve under time, in probabilistic system, there is some indeterminacy in its future evolution

19 2.1 Probabilistic Systems 12 described by the probability distributions. There are many possibilities the process might go to, but some paths are more probable and others are less. The time evolution of a probabilistic system develops each state x i into distribution x i p 1i [x 1 ] + p 2i [x 2 ] p ni [x n ], (2.2) such that p 1i + p 2i p ni = 1 for each i. In the notation 2.2, p ji is the probability that the system x i into x j. Thus a distribution p 1 [x 1 ] + p 2 [x 2 ] p n [x n ] (2.3) evolves into p 1 (p 11 [x 1 ] p n1 [x n ]) p n (p 1n [x 1 ] + p 2i [x 2 ] p nn [x n ]) = = (p 11 p p 1n p n )[x 1 ] (p n1 p p nn p n )[x n ] = = p 1[x 1 ] p n[x n ], where p i = p 1i p p 1i p n. Therefore the probabilities p i and p i are related by p 11 p p 1n p 21 p p 2n p 1 p 2. = p 1 p 2.. (2.4) p n1 p n2... p nn p n p n The matrix in the equation 2.4 is a stochastic matrix, also called Markov matrix, it has non-negative entries and p 1i + p 2i p ni = 1 for each i, which guaranties that p 1 + p p n = p 1 + p p n. Definition A real (n x n) matrix A = [a ij ] is called a Markov matrix or stochastic matrix if a ij > 0 for 1 i, j n n a ij = 1 for 1 j n. i=0 A probabilistic system with a time evaluation described above is called a Markov chain.

20 2.2 Quantum Mechanics Quantum Mechanics Here, we give a brief introduction to quantum mechanics. The term "quantum mechanics" is used for the mathematical structure describing "quantum physics". Quantum mechanics was born in the beginnings of the twentieth century, when experiments on atoms and radiations could not be fully explained by classical physics even by using the Markov chain described in the previous section. At first, we introduce the formalism of quantum mechanics in a basic form on state vectors and we assume that the quantum system is a finite - dimensional Superposition The quantum mechanical description of a physical system looks similar to description of a probabilistic system in 2.1. However, they differ essentially. In quantum mechanics, a state of an n-level system is described as a unit-length vector in an n-dimensional complex vector space H n called Hilbert space (see Subsection 2.2.2). H n is called the state space of the system. To define a quantum state we choose an orthonormal basis { x 1, x 2,..., x n } 1 for the Hilbert space H n. Definition A pure quantum state φ is a superposition of a classical states, written φ = α 1 x 1 + α 2 x α n x n, (2.5) where α i are complex numbers called amplitudes (with respect to the chosen basis) and α α α n 2 = 1, where α j 2 is the squared norm of the corresponding amplitude α j ( ai + b = a 2 + b 2 ). The quantum state 2.5 can be also seen as the n-dimensional column vector α 1 α 2... α n. (2.6) Example A two-level quantum system can be used to represent a bit. Such a system is called a quantum bit or qubit. The system has an orthonormal basis { 0, 1 }. A general state of the system is represented by α α 1 1, where α α 2 2 = 1. The probability to see the system to have the property 0 is α 0 2, and 1 - α Notation like x is standard notation for states in quantum mechanics and is called "ket" notation or Dirac notation.

21 2.2 Quantum Mechanics Hilbert Space Hilbert space is a mathematical framework suitable for describing the concepts and principles of quantum system. In this subsection, we will define Hilbert space. Definition An inner-product space H is a complex vector space, equipped with an inner product : H H C satisfying the following axioms for any vectors φ, ψ, φ 1, φ 2 H, and any c 1, c 2 C. 1. φ ψ = ψ φ 2, 2. ψ ψ 0 and ψ ψ = 0 if and only if ψ = 0, 3. ψ c 1 φ 1 + c 2 φ 2 = c 1 ψ φ 1 + c 2 ψ φ 2. The inner product introduces on H the norm or length ψ = ψ ψ and the metric (Euclidean distance) dist(φ, ψ) = φ ψ. Definition An inner-product vector space H is called complete if for each vector sequence ψ i such that lim m,n ψ m ψ n = 0, there exists a vector ψ H such that lim n ψ n ψ = 0. A complete inner-product space is called a Hilbert space. An n-dimensional complex vector Hilbert space is denoted by H n or C n. A vector ψ of an n-dimensional Hilbert space is denoted by ψ, and is referred as a ket-vector, and it can be seen as an n-dimensional column vector 2.6. The ψ is referred to as bra-vector and can be seen as an n-dimensional row vector ( α 1 α 2... α n ), (2.7) where α i are complex numbers. The transformation ψ ψ corresponds to transposition and conjunction. The inner product ψ φ is then "row vector column vector" product, which produces a complex number as output. The outer product ψ φ is an n n matrix - "column vector row vector" product. Example Let us consider a two states of the H 2 : ψ 1 = ( ) β 0 β 1. The inner product of ψ 1 and ψ 2 is ( ) ( ) β ψ 1 ψ 2 = α0 α1 0 = α0β 0 + α1β 1 2 Notation z denotes the complex conjugate of the complex number z: (a + bi) = a bi. β 1 ( α 0 α 1 ) and ψ 2 =

22 2.2 Quantum Mechanics 15 and the outer product is ψ 1 ψ 2 = ( α 0 α 1 ) ( β 0 β 1 ( ) = α 0 β 0 α 0 β 1 α 1 β 0 α 1 β 1 ) Observables and Measurement In order to extract quantum information from a quantum system, we have to observe the system to perform a measurement. At first, we consider a measurement in the computational basis. Suppose, that a quantum system is in a state φ = α 1 x 1 + α 2 x α n x n. We cannot "see" a superposition itself, but only classical states. The basis state x j is observed with probability of α j 2, which is the squared norm of the corresponding amplitude α j ( ai + b = a 2 + b 2 ). Observing φ collapses the quantum state φ to the classic state x j and all the "information" that might have been contained in the α j is gone. We may generalize the measurement as follows: Definition An observable E = {E 1, E 2,..., E m } (m n) is a collection of mutually orthogonal subspaces of the Hilbert space H n such that H n = E 1 E 2... E m. We equip the subspaces E i with distinct real number "labels" θ 1, θ 2,..., θ m. Each vector x H n can be decomposed in a unique way as x = x 1 + x x m such that x i E i. Instead of observing subspaces E i, we can talk about observing the labels θ i : by observing E, value θ i will be seen with a probability of x i 2. Example The observable E can be defined as E = {E 1, E 2,..., E n }, where each E i is the one-dimensional subspace spanned to x i. Now, we equip the subspaces E i with the label i. If the subspace is in the state 2.5, the value i is observed with a probability of α i x i = α i 2. Another way of viewing the observables is using projectors. The measurement is described by projectors P 1, P 2,..., P m (m n), which sum to identity. The projectors are orthogonal (P i P j = 0, if i j). The projector P j projects on the subspace E j of the Hilbert space H n, and every vector x H n can be decomposed in a unique way as

23 2.2 Quantum Mechanics 16 x = x 1 + x x m such that x j = P j x E j. We will get outcome θ j with probability x j 2 = T r(p j x x ) and the state will collapse to the new state x = x j x j = P j x P j x. Example If we consider the previous example using projectors, then n = m and P i = x i x i. Applying our measurement to ψ we will get output i with probability P i ψ 2 = α i x i 2 = α i 2 and the state collapses to α i x i = α i x α i x i α i i Unitary Evolution We have measured a quantum state, but what about the time evaluation of the quantum system? For quantum systems, the time evaluation of probabilistic systems via Markov matrices is replaced by matrices with complex number entries that preserve the constraint n i=1 α i 2 = 1. Thus, the quantum system state φ = α 1 x 1 + α 2 x α n x n evolves into the state φ = α 1 x 1 + α 2 x α n x n, where amplitudes α 1, α 2,..., α n and α 1, α 2,..., α n are related by α 11 α α 1n α 21 α α 2n α 1 α 2. = α 1 α 2. (2.8) α n1 α n2... α nn p n α n and n i=1 α i 2 = n i=1 α i 2 = 1. Thus, the quantum systems time evaluation should be unitary transformation. Definition A complex matrix U is called unitary if UU = U U = I n, where I n is identity matrix in n dimensions and U is conjugate transpose of U. As unitary transformation always has inverse, it follows that time evaluation (nonmeasurement) must be reversible. The measurement is not reversible. Example Let us consider a quantum coin flipping. We have a quantum system with

24 2.2 Quantum Mechanics 17 two basis states 0 and 1 and time evolution Notice that if we start either in state 0 or 1 after the first toss we can observe both options with probability 1, but if we do not observe the system with the second toss we 2 return to the starting state. Definition A Hermitian matrix or self-adjoint matrix is a square matrix with complex entries which is equal to its own conjugate transpose (A = A ) Mixed States and Density Matrices Pure states are fundamental objects for quantum mechanics in the sense that evolution of any closed system can be seen as a unitary evolution of pure states. However, to deal with opened and composed quantum systems the concept of mixed state is important. Definition A probability distribution {(p i, φ i ) 1 i n} on pure states {φ i } n i=1, with probabilities 0 < p i 1, where n i=1 p i = 1, is called a mixed state or probabilistic mixture, and denoted by [ψ = {(p i, φ i ) 1 i n} or [ψ = p 1 φ 1... p n φ n = n p i φ i. i=1 The result of the measurement of a pure state ψ = n i=1 α i φ i with respect to the observable given by an orthonormal basis {φ} n i=1 can be consider as the mixed state [ψ = n α i 2 φ i. i=1 Definition To each mixed state [ψ = n i=1 p iφ i corresponds a density matrix or density operator n ρ [ψ = p i φ i φ i. i=1 However, the density operator of a mixed state does not capture all the information about a mixed state. Different mixed states can have the same density operator. For

25 2.2 Quantum Mechanics 18 example, if [ψ 1 = and [ψ 2 = 1 2 ( ) 1 ( 0 1 ) 2 the corresponding density matrix for both mixed states is ρ ψ1 = ρ ψ2 = Now, we list some important properties of density matrices ρ = x x : matrices ρ have a unit trace 3. ρ is Hermitian. Eigenvalues of ρ are non-negative. ρ = ρ 2 if and only if ρ is a density matrix of a pure state. If ρ is a density matrix of mixed state then ρ 2 < ρ and T r(ρ 2 ) < 1. If ρ is a density matrix of a pure state then it has one eigenvalue equal to 1 and all other eigenvalues equal to 0. Evaluation of density operator For a pure state ψ, the evaluation can be described with unitary transformation as U ψ, where U is a unitary matrix. For the mixed state, if the system was in the state ψ i with probability p i then after the evaluation it will be in the state U ψ i with probability p i. Thus, the evaluation of the density operator can be described by the equation ρ = i p i ψ i ψ i U i p i U ψ i ψ i U = UρU. (2.9) Measurement Suppose we perform measurements with projectors P 1, P 2,..., P m. If the initial state is ψ i, then the probability of obtaining the result j is p(j i) = T r(p j ψ i ψ i ). (2.10) 3 Trace of the matrix ρ denoted by T r(ρ) is the sum of the diagonal elements.

26 2.3 Quantum Finite State Automata 19 Using the law of total probability the probability of getting result j is p(j) = i p(j i)p i = i T r(p j ψ i ψ i ) = T r(p j ρ) (2.11) Using the techniques of the probability theory we can obtain that the density operator after obtaining measurement j is ρ j = P jρp j T r(p j ρ). (2.12) 2.3 Quantum Finite State Automata A classical finite state automaton servers as a basic model of a classical finite size machine. Similarly, a quantum finite state automaton can be seen as a basic quantum model of finite state quantum machines Models of Classical Finite State Automata Different models of finite state automata have been developed and investigated in the classical computation. We consider the very basic notation of finite state automata - deterministic finite state automata, non-deterministic finite state automata, and probabilistic finite state automata. Definition A deterministic finite state automaton (DFA) is a tuple A = (Q, Σ, δ, q 0, Q a ) where: Q is a finite set of states. Σ is a finite set of input symbols. It is said to be alphabet of the automaton. δ : Q Σ Q is a transition function. q 0 Q is an initial state (the state of the automaton when no input has been processed). Q a Q is a set of accepting states. An automaton reads a finite string of input letters a 1 a 2...a n (a i Σ), which is called an input word. Set of all words is denoted by Σ. Sometimes a special character is used to denote the end of the word. The automaton starts the computation at the initial state q 0 and performs the transition function corresponding to the current state and the current input symbol.

27 2.3 Quantum Finite State Automata 20 q0 b a q1 a b a,b q2 Figure 2.1 The deterministic finite state automaton A 1. Definition A run of the automaton A = (Q, Σ, δ, q 0, Q a ) on the input word a 1 a 2...a n Σ is a sequence of states q 0 q 1...q n, where q i Q and q i = δ(q i 1, a i ). (q n is the final state of the run.) Definition We say, the automaton A = (Q, Σ, δ, q 0, Q a ) accepts or recognizes an input word a 1 a 2...a n if q n Q a. Definition We say, the automaton A = (Q, Σ, δ, q 0, Q a ) recognizes the language L(A), if L(A) contains all the words recognized by the automaton A. Example Let us consider the automaton A 1 = (Q, Σ, δ, q 0, Q a ), where Q = {q 0, q 1, q 2 } Σ = {a, b} δ can be described by a b q 0 q 1 q 0 q 1 q 1 q 2 q 2 q 1 q 1 q 0 is the initial state Q a = {q 1 } The automaton can also be characterized using the state diagram (see the figure 2.1), where it has three states labelled q 0, q 1, and q 2. The initial state is indicated by the arrow pointing to it from nowhere, for the current example - it is the state q 0. The accepting states are indicated by double circle (in the example - {q 1 }). The arrow from the state q i to the state q j with label a k denotes transition q j = δ(q i, a k ). In our example, the automaton A 1 recognizes language L 2 = { w w has at least 1 a and even number of b s following the last a}.

28 2.3 Quantum Finite State Automata 21 b b q0 a q1 a a q2 b Figure 2.2 The group finite state automaton recognizing the language a 3n. Besides deterministic finite state automaton there are also other variations in different components of automata (input, transition, acceptance). For example, an automata can accept finite or infinite words (the infinite words will be consider later in the section 2.6), or tree structures. Let us consider three other variations of finite state automata - group finite state automata, non-deterministic finite state automata, and probabilistic finite state automata. Definition A group finite state automaton (GFA) is a deterministic finite automaton A QF A = (Q, Σ, δ, q 0, Q a ) with the restriction that for every state q Q and every input symbol σ Σ there exists exactly one state q Q such that δ(q, σ) = q. In other words, δ is a complete one-to-one function and the automaton derived from A GF A by reversing all transitions is deterministic. The automaton in the figure 2.1 is not a group automaton as there are three states q 0, q 1, and q 2 from which we can reach the state q 1 with input letter a. In the figure 2.2, a group finite state automata recognizing the language {a 3n n 0} is displayed. Definition A non-deterministic finite state automaton (NFA) is a tuple A NF A = (Q, Σ, δ, q 0, Q a ) where: Q is a finite set of states. Σ is a finite set of input symbols. It is said to be alphabet of the automaton. δ : Q (Σ {ɛ}) P (Q) is a transition function, where P (Q) denotes the power set 4 of Q. q 0 Q is an initial state (the state of the automaton when no input has been processed). Q a Q is a set of accepting states. 4 P (Q) def = {X : X Q} is the power set of Q.